Triangulations of the `magic manifold' and families of census knots

TL;DR

This work develops five explicit ideal triangulations of the 3-cusped hyperbolic manifold $M_3$ (the magic manifold) that are compatible with Dehn-filling techniques using layered solid tori and layered chains. By applying these constructions, the authors realize low-complexity triangulations for all partial fillings of $M_3$ and recover minimal triangulations for 229 census knots, organizing 221 of them into 42 twisting families with provable upper bounds on triangulation size. They provide a systematic framework based on the Farey diagram, boundary cusps, and Hopf-model geometry to count tetrahedra and identify minimal triangulations, offering both concrete triangulations and conjectured minimality within each family. The results establish a bridge between hyperbolic 3-manifold triangulations and census knot complexity, with broad implications for algorithmic efficiency in 3-manifold topology and for understanding how infinite families of knots inherit bounded triangulation complexity. The methodology enables scalable exploration of Dehn fillings and their minimal realizations, potentially guiding future classifications beyond census knots.

Abstract

We describe five ideal triangulations of the 3-cusped hyperbolic `magic manifold' that are each compatible with well-established techniques for triangulating Dehn fillings. Using these techniques, we construct low-complexity triangulations for all partial fillings of the magic manifold, and in particular, recover minimal triangulations for 229 of the hyperbolic census knots. Along the way, these census knots are sorted into 42 families related by twisting that can be extended indefinitely, with each member of each infinite family inheriting an upper bound on its triangulation complexity. These triangulations are conjectured to be minimal for all 42 families.

Figures (17)

• Figure 1: An isotopy of 3CL from its chain diagram to the diagram used in Figure \ref{['fig:MM-diagrams']}.
• Figure 2: Left: The alternative diagram of 3CL from Figure \ref{['fig:3CL-isotopy']}. Centre: A projection of the left diagram onto $T_0$, which can also be seen as a top-down view of the Hopf model on the right. Right: The (truncated) Hopf model $\mathbf{H}\xspace_M$ for $M_3$.
• Figure 3: The 2-component parent link $M_3(-3)$ responsible for the satellite knots we encounter.
• Figure 4: A permissible boundary that a layered chain may be attached to. We say that the boundary on the left has positive diagonals, while the boundary on the right has negative diagonals.
• Figure 5: Instructions for closing a layered chain by folding. This shows the case where the permissible boundary has positive diagonals, and the case with negative diagonals is the same up to reflection.

Theorems & Definitions (19)

• Theorem 2.1: Martelli-Petronio Martelli-Petronio
• Remark 2.2
• proof
• Lemma 2.4
• proof
• Definition 2.5
• Remark 3.1
• Definition 4.1
• Definition 4.2
• Definition 5.2